Copied to
clipboard

## G = D26⋊3Q8order 416 = 25·13

### 3rd semidirect product of D26 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — D26⋊3Q8
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — D26⋊3Q8
 Lower central C13 — C2×C26 — D26⋊3Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for D263Q8
G = < a,b,c,d | a26=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd-1=c-1 >

Subgroups: 472 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, D13 [×2], C26 [×3], C22⋊Q8, Dic13 [×3], C52 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C4×D13 [×2], C2×Dic13, C2×Dic13 [×2], C2×C52, C2×C52 [×2], Q8×C13 [×2], C22×D13, C26.D4 [×2], C523C4, D26⋊C4 [×2], C2×C4×D13, Q8×C26, D263Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26 [×3], C13⋊D4 [×2], C22×D13, Q8×D13, D52⋊C2, C2×C13⋊D4, D263Q8

Smallest permutation representation of D263Q8
On 208 points
Generators in S208
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(105 144)(106 143)(107 142)(108 141)(109 140)(110 139)(111 138)(112 137)(113 136)(114 135)(115 134)(116 133)(117 132)(118 131)(119 156)(120 155)(121 154)(122 153)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(157 192)(158 191)(159 190)(160 189)(161 188)(162 187)(163 186)(164 185)(165 184)(166 183)(167 208)(168 207)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(177 198)(178 197)(179 196)(180 195)(181 194)(182 193)
(1 205 59 158)(2 206 60 159)(3 207 61 160)(4 208 62 161)(5 183 63 162)(6 184 64 163)(7 185 65 164)(8 186 66 165)(9 187 67 166)(10 188 68 167)(11 189 69 168)(12 190 70 169)(13 191 71 170)(14 192 72 171)(15 193 73 172)(16 194 74 173)(17 195 75 174)(18 196 76 175)(19 197 77 176)(20 198 78 177)(21 199 53 178)(22 200 54 179)(23 201 55 180)(24 202 56 181)(25 203 57 182)(26 204 58 157)(27 140 82 120)(28 141 83 121)(29 142 84 122)(30 143 85 123)(31 144 86 124)(32 145 87 125)(33 146 88 126)(34 147 89 127)(35 148 90 128)(36 149 91 129)(37 150 92 130)(38 151 93 105)(39 152 94 106)(40 153 95 107)(41 154 96 108)(42 155 97 109)(43 156 98 110)(44 131 99 111)(45 132 100 112)(46 133 101 113)(47 134 102 114)(48 135 103 115)(49 136 104 116)(50 137 79 117)(51 138 80 118)(52 139 81 119)
(1 135 59 115)(2 136 60 116)(3 137 61 117)(4 138 62 118)(5 139 63 119)(6 140 64 120)(7 141 65 121)(8 142 66 122)(9 143 67 123)(10 144 68 124)(11 145 69 125)(12 146 70 126)(13 147 71 127)(14 148 72 128)(15 149 73 129)(16 150 74 130)(17 151 75 105)(18 152 76 106)(19 153 77 107)(20 154 78 108)(21 155 53 109)(22 156 54 110)(23 131 55 111)(24 132 56 112)(25 133 57 113)(26 134 58 114)(27 163 82 184)(28 164 83 185)(29 165 84 186)(30 166 85 187)(31 167 86 188)(32 168 87 189)(33 169 88 190)(34 170 89 191)(35 171 90 192)(36 172 91 193)(37 173 92 194)(38 174 93 195)(39 175 94 196)(40 176 95 197)(41 177 96 198)(42 178 97 199)(43 179 98 200)(44 180 99 201)(45 181 100 202)(46 182 101 203)(47 157 102 204)(48 158 103 205)(49 159 104 206)(50 160 79 207)(51 161 80 208)(52 162 81 183)```

`G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,156)(120,155)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(157,192)(158,191)(159,190)(160,189)(161,188)(162,187)(163,186)(164,185)(165,184)(166,183)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193), (1,205,59,158)(2,206,60,159)(3,207,61,160)(4,208,62,161)(5,183,63,162)(6,184,64,163)(7,185,65,164)(8,186,66,165)(9,187,67,166)(10,188,68,167)(11,189,69,168)(12,190,70,169)(13,191,71,170)(14,192,72,171)(15,193,73,172)(16,194,74,173)(17,195,75,174)(18,196,76,175)(19,197,77,176)(20,198,78,177)(21,199,53,178)(22,200,54,179)(23,201,55,180)(24,202,56,181)(25,203,57,182)(26,204,58,157)(27,140,82,120)(28,141,83,121)(29,142,84,122)(30,143,85,123)(31,144,86,124)(32,145,87,125)(33,146,88,126)(34,147,89,127)(35,148,90,128)(36,149,91,129)(37,150,92,130)(38,151,93,105)(39,152,94,106)(40,153,95,107)(41,154,96,108)(42,155,97,109)(43,156,98,110)(44,131,99,111)(45,132,100,112)(46,133,101,113)(47,134,102,114)(48,135,103,115)(49,136,104,116)(50,137,79,117)(51,138,80,118)(52,139,81,119), (1,135,59,115)(2,136,60,116)(3,137,61,117)(4,138,62,118)(5,139,63,119)(6,140,64,120)(7,141,65,121)(8,142,66,122)(9,143,67,123)(10,144,68,124)(11,145,69,125)(12,146,70,126)(13,147,71,127)(14,148,72,128)(15,149,73,129)(16,150,74,130)(17,151,75,105)(18,152,76,106)(19,153,77,107)(20,154,78,108)(21,155,53,109)(22,156,54,110)(23,131,55,111)(24,132,56,112)(25,133,57,113)(26,134,58,114)(27,163,82,184)(28,164,83,185)(29,165,84,186)(30,166,85,187)(31,167,86,188)(32,168,87,189)(33,169,88,190)(34,170,89,191)(35,171,90,192)(36,172,91,193)(37,173,92,194)(38,174,93,195)(39,175,94,196)(40,176,95,197)(41,177,96,198)(42,178,97,199)(43,179,98,200)(44,180,99,201)(45,181,100,202)(46,182,101,203)(47,157,102,204)(48,158,103,205)(49,159,104,206)(50,160,79,207)(51,161,80,208)(52,162,81,183)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,156)(120,155)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(157,192)(158,191)(159,190)(160,189)(161,188)(162,187)(163,186)(164,185)(165,184)(166,183)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193), (1,205,59,158)(2,206,60,159)(3,207,61,160)(4,208,62,161)(5,183,63,162)(6,184,64,163)(7,185,65,164)(8,186,66,165)(9,187,67,166)(10,188,68,167)(11,189,69,168)(12,190,70,169)(13,191,71,170)(14,192,72,171)(15,193,73,172)(16,194,74,173)(17,195,75,174)(18,196,76,175)(19,197,77,176)(20,198,78,177)(21,199,53,178)(22,200,54,179)(23,201,55,180)(24,202,56,181)(25,203,57,182)(26,204,58,157)(27,140,82,120)(28,141,83,121)(29,142,84,122)(30,143,85,123)(31,144,86,124)(32,145,87,125)(33,146,88,126)(34,147,89,127)(35,148,90,128)(36,149,91,129)(37,150,92,130)(38,151,93,105)(39,152,94,106)(40,153,95,107)(41,154,96,108)(42,155,97,109)(43,156,98,110)(44,131,99,111)(45,132,100,112)(46,133,101,113)(47,134,102,114)(48,135,103,115)(49,136,104,116)(50,137,79,117)(51,138,80,118)(52,139,81,119), (1,135,59,115)(2,136,60,116)(3,137,61,117)(4,138,62,118)(5,139,63,119)(6,140,64,120)(7,141,65,121)(8,142,66,122)(9,143,67,123)(10,144,68,124)(11,145,69,125)(12,146,70,126)(13,147,71,127)(14,148,72,128)(15,149,73,129)(16,150,74,130)(17,151,75,105)(18,152,76,106)(19,153,77,107)(20,154,78,108)(21,155,53,109)(22,156,54,110)(23,131,55,111)(24,132,56,112)(25,133,57,113)(26,134,58,114)(27,163,82,184)(28,164,83,185)(29,165,84,186)(30,166,85,187)(31,167,86,188)(32,168,87,189)(33,169,88,190)(34,170,89,191)(35,171,90,192)(36,172,91,193)(37,173,92,194)(38,174,93,195)(39,175,94,196)(40,176,95,197)(41,177,96,198)(42,178,97,199)(43,179,98,200)(44,180,99,201)(45,181,100,202)(46,182,101,203)(47,157,102,204)(48,158,103,205)(49,159,104,206)(50,160,79,207)(51,161,80,208)(52,162,81,183) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,59),(27,84),(28,83),(29,82),(30,81),(31,80),(32,79),(33,104),(34,103),(35,102),(36,101),(37,100),(38,99),(39,98),(40,97),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(105,144),(106,143),(107,142),(108,141),(109,140),(110,139),(111,138),(112,137),(113,136),(114,135),(115,134),(116,133),(117,132),(118,131),(119,156),(120,155),(121,154),(122,153),(123,152),(124,151),(125,150),(126,149),(127,148),(128,147),(129,146),(130,145),(157,192),(158,191),(159,190),(160,189),(161,188),(162,187),(163,186),(164,185),(165,184),(166,183),(167,208),(168,207),(169,206),(170,205),(171,204),(172,203),(173,202),(174,201),(175,200),(176,199),(177,198),(178,197),(179,196),(180,195),(181,194),(182,193)], [(1,205,59,158),(2,206,60,159),(3,207,61,160),(4,208,62,161),(5,183,63,162),(6,184,64,163),(7,185,65,164),(8,186,66,165),(9,187,67,166),(10,188,68,167),(11,189,69,168),(12,190,70,169),(13,191,71,170),(14,192,72,171),(15,193,73,172),(16,194,74,173),(17,195,75,174),(18,196,76,175),(19,197,77,176),(20,198,78,177),(21,199,53,178),(22,200,54,179),(23,201,55,180),(24,202,56,181),(25,203,57,182),(26,204,58,157),(27,140,82,120),(28,141,83,121),(29,142,84,122),(30,143,85,123),(31,144,86,124),(32,145,87,125),(33,146,88,126),(34,147,89,127),(35,148,90,128),(36,149,91,129),(37,150,92,130),(38,151,93,105),(39,152,94,106),(40,153,95,107),(41,154,96,108),(42,155,97,109),(43,156,98,110),(44,131,99,111),(45,132,100,112),(46,133,101,113),(47,134,102,114),(48,135,103,115),(49,136,104,116),(50,137,79,117),(51,138,80,118),(52,139,81,119)], [(1,135,59,115),(2,136,60,116),(3,137,61,117),(4,138,62,118),(5,139,63,119),(6,140,64,120),(7,141,65,121),(8,142,66,122),(9,143,67,123),(10,144,68,124),(11,145,69,125),(12,146,70,126),(13,147,71,127),(14,148,72,128),(15,149,73,129),(16,150,74,130),(17,151,75,105),(18,152,76,106),(19,153,77,107),(20,154,78,108),(21,155,53,109),(22,156,54,110),(23,131,55,111),(24,132,56,112),(25,133,57,113),(26,134,58,114),(27,163,82,184),(28,164,83,185),(29,165,84,186),(30,166,85,187),(31,167,86,188),(32,168,87,189),(33,169,88,190),(34,170,89,191),(35,171,90,192),(36,172,91,193),(37,173,92,194),(38,174,93,195),(39,175,94,196),(40,176,95,197),(41,177,96,198),(42,178,97,199),(43,179,98,200),(44,180,99,201),(45,181,100,202),(46,182,101,203),(47,157,102,204),(48,158,103,205),(49,159,104,206),(50,160,79,207),(51,161,80,208),(52,162,81,183)])`

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 26 26 2 2 4 4 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 D13 D26 C13⋊D4 Q8×D13 D52⋊C2 kernel D26⋊3Q8 C26.D4 C52⋊3C4 D26⋊C4 C2×C4×D13 Q8×C26 C52 D26 C26 C2×Q8 C2×C4 C4 C2 C2 # reps 1 2 1 2 1 1 2 2 2 6 18 24 6 6

Matrix representation of D263Q8 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 33 4 0 0 31 7
,
 1 0 0 0 0 52 0 0 0 0 1 0 0 0 20 52
,
 0 1 0 0 52 0 0 0 0 0 25 24 0 0 27 28
,
 30 0 0 0 0 23 0 0 0 0 52 0 0 0 0 52
`G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,33,31,0,0,4,7],[1,0,0,0,0,52,0,0,0,0,1,20,0,0,0,52],[0,52,0,0,1,0,0,0,0,0,25,27,0,0,24,28],[30,0,0,0,0,23,0,0,0,0,52,0,0,0,0,52] >;`

D263Q8 in GAP, Magma, Sage, TeX

`D_{26}\rtimes_3Q_8`
`% in TeX`

`G:=Group("D26:3Q8");`
`// GroupNames label`

`G:=SmallGroup(416,167);`
`// by ID`

`G=gap.SmallGroup(416,167);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,86,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
×
𝔽